On perturbation methods in nonlinear stability theory

Abstract

This paper reconsiders formal expansion methods for the analysis of the nonlinear properties of a modal disturbance. A survey is given of the various types of expansions based on different assumptions, and their range and shortcomings are discussed. By introducing a well-defined amplitude, Watson's expansion in a time-dependent amplitude is developed into a rational method for uniquely determining Landau constants of arbitrary order. Complementary to the common orthogonality condition for points at the neutral curve, an alternative definition of the Landau constants is given for points in the unstable domain. The method is not restricted to small amplification rates but is invalid in the stable domain. The method of Reynolds and Potter for a direct attack on equilibrium states is extended into a class of rational methods. The methods in this class agree to within a rearrangement of the infinite expansion series but differ in their respective range of validity.